How many unique simple forms of rational numbers are there of the form
p/q, where p and q are non-zero whole numbers less than or equal to n?
For example, 1/2 and 2/4 have the same simple form, so they are not
considered unique. The answer should be a function of n.

Find integer solutions of the equation x^3 + y^3 = 31z^3. I know the
fundamental solution is (137, -65, 42), but I want to have all the
values positive. I know also that there is an arithmetic procedure
(doubling in the group) to obtain further solutions from the
fundamental one, but I do not know the details of this procedure.

A student wonders if a u-substitution would solve a troublesome Diophantine
equation. Doctor Vogler takes up the struggle with a false start of his own, before
noticing helpful patterns from a computational attack and walking the student through
the thought process behind his strategy.

I need to know how to get positive integer solutions of two
Diophantine equations having three variables. For example: 2x + 3y +
7z = 32 ; 3x + 4y - z = 19. (Give the positive set of triples for the
above equations.)

I would like to find out how to develop the parameters of a cubic
parabola in general so that I can implement an integer factorization
method. I would also like to know how to add points such as P+P and
P+Q to such a curve.